The general solution of $\left(1+e^{\frac{x}{y}}\right) d x+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) d y=0$ is

If the general solution of $\frac{dy}{dx} = \frac{y^2}{xy - y^2 - x^2}$ is $\tan^{-1}\left(\frac{y}{x}\right) = f(y) + C$,then $f(e^3) = $

The general solution of the differential equation $\left(x \sin \frac{y}{x}\right) dy = \left(y \sin \frac{y}{x} - x\right) dx$ is

If $y=y(x)$ is the solution of $x \frac{d y}{d x}=y+x e^{-\left(\frac{y}{x}\right)}$ with $y(1)=\log e$,then find $y(e)$.

If the solution curve $y=y(x)$ of the differential equation $y^{2} dx + (x^{2} - xy + y^{2}) dy = 0$ passes through the point $(1, 1)$ and intersects the line $y = \sqrt{3}x$ at the point $(\alpha, \sqrt{3}\alpha)$,then the value of $\log_{e}(\sqrt{3}\alpha)$ is equal to

Which of the following is a homogeneous differential equation?